Asymptotic analysis of the mixed pseudo-parabolic-Kirchhoff equation with nonstandard growth condition

In this paper, a class of mixed pseudo-parabolic-Kirchhoff equation is considered. We focus on the interaction between the nonlocal diffusion and the nonstandard growth condition. More specifically, we establish a threshold result for the global solutions and finite time blow-up of the weak solution for low and critical initial energy. Meanwhile, we give the upper and lower bound estimates of blow-up time for the blow-up solution. And then we investigate the convergent relation between the global solutions and the ground-state solutions. Finally, we give some conditions to ensure the global existence or blow-up of solutions for the high initial energy.